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In this note, we show that the solution to the Dirichlet problem for the minimal surface system in any codimension is unique in the space of distance-decreasing maps. This follows as a corollary of the following stability theorem: if a minimal submanifold $\sum$  is the graph of a (strictly) distance-decreasing map, then  $\sum$ is (strictly) stable. It is known that a minimal graph of codimension one is stable without assuming the distance-decreasing condition. We give another criterion for the stability in terms of the two-Jacobians of the map which in particular covers the codimension one case. All theorems are proved in the more general setting for minimal maps between Riemannian manifolds. The complete statements of the results appear in Theorem 3.1, Theorem 3.2, and Theorem 4.1.

We study weakly trapped submanifolds of codimension two in a standard static spacetime. In this setting, we apply some generalized maximum principles in order to investigate the geometry of these trapped submanifolds. For instance, we establish sufficient conditions to guarantee that such a spacelike submanifold must be a hypersurface of the Riemannian base of the ambient spacetime. As a consequence, we prove that there do not exist n-dimensional compact (without boundary) trapped submanifolds immersed in an (n+2)-dimensional standard static spacetime. Such a nonexistence result was originally obtained for stationary spacetimes by Mars and Senovilla [20]. Furthermore, we also investigate parabolic weakly trapped submanifolds immersed in a standard static spacetime.

The limiting behavior of the normalized KÄahler-Ricci °ow for manifolds with positive ¯rst Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi Kenergy is bounded from below and if the lowest positive eigenvalue of the ¹@y ¹@ operator on smooth vector ¯elds is bounded away from0 along the °ow, then the metrics converge exponentially fast in C1 to a KÄahler-Einstein metric.

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